Cracking the Cubic:
Cardano, Controversy, and Creasing
Alissa S. Crans
Loyola Marymount University
MAA MD-DC-VA Spring Meeting
Stevenson University
April 14, 2012
These images are from the Wikipedia articles on Niccolò Fontana Tartaglia and Gerolamo Cardano. Both images belong to the public domain.
Wednesday, April 18, 2012
Quadratic Equation
A brief history...
• 400 BC
Babylonians
• 300 BC
Euclid
323 - 283 BC
This image is from the website entry for Euclid from the MacTutor History of Mathematics. It belongs to the public domain.
Wednesday, April 18, 2012
Quadratic Equation
•
600 AD
Brahmagupta
To the absolute number multiplied by four
times the [coefficient of the] square, add the
square of the [coefficient of the] middle term;
the square root of the same, less the [coefficient
of the] middle term, being divided by twice the
[coefficient of the] square is the value.
Brahmasphutasiddhanta
Colebook translation, 1817, pg 346
598 - 668 BC
ax2 + bx = c
x=
√
4ac + b2 − b
2a
This image is from the website entry for Brahmagupta from the The Story of Mathematics. It belongs to the public domain.
Wednesday, April 18, 2012
Quadratic Equation
• 800 AD
al-Khwarizmi
• 12th cent
bar Hiyya (Savasorda)
Liber embadorum
• 13th cent
Yang Hui
780 - 850
This image is from the website entry for al-Khwarizmi from The Story of Mathematics. It belongs to the public domain.
Wednesday, April 18, 2012
Luca Pacioli
1445 - 1509
Summa de arithmetica, geometrica, proportioni
et proportionalita (1494)
This image is from the Wikipedia article on Luca Pacioli. It belongs to the public domain.
Wednesday, April 18, 2012
Cubic Equation
Challenge: Solve the equation
ax3 + bx2 + cx + d = 0
The quest for the solution to the cubic begins!
Enter Scipione del Ferro...
Wednesday, April 18, 2012
Scipione del Ferro
• 1465 - 1526, Italian
• Chair of math dept at University of Bologna
3 + mx = n
First
to
solve
depressed
cubic:
x
•
• Kept formula secret!
• Revealed method to student Antonio Fior on
deathbed
Wednesday, April 18, 2012
Nicolo of Brescia (Tartaglia)
• 1500 - 1557, Italian
3 + mx2 = n
Feb
13,
1535
solved
x
•
• Won challenge!
This image is from the Wikipedia article on Niccolò Fontana Tartaglia. It belongs to the public domain.
Wednesday, April 18, 2012
Girolamo Cardano
• 1501 - 1576, Italian
• Numerous ailments when young
• Became a physician
• Wrote treatise on probability
• Brought Tartaglia to Milan to
learn secret of the cubic
This image is from the Wikipedia article on Gerolamo Cardano. It belongs to the public domain.
Wednesday, April 18, 2012
The (encoded) solution!
When the cube and things together
Are equal to some discreet number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing
Wednesday, April 18, 2012
The (encoded) solution!
In the second of these acts,
When the cube remains alone,
You will observe these other agreements:
You will at once divide the number into two parts
So that the one times the other produces clearly
The cube of the third of the things exactly.
Then of these two parts, as a habitual rule,
You will take the cube roots added together,
And this sum will be your thought.
Wednesday, April 18, 2012
The (encoded) solution!
The third of these calculations of ours
Is solved with the second if you take good care,
As in their nature they are almost matched.
These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea.
Wednesday, April 18, 2012
The (encoded) solution!
This verse speaks so clearly that, without any
other example, I believe that your Excellency will
understand everything. - Tartaglia
I swear to you, by God's holy Gospels, and as a true
man of honour, not only never to publish your
discoveries, if you teach me them, but I also promise
you, and I pledge my faith as a true Christian, to note
them down in code, so that after my death no one will
be able to understand them. - Cardano
Wednesday, April 18, 2012
Lodovico Ferrari
• 1522 - 1565, Italian
• Started out as Cardano’s servant
• Quickly became colleagues
• Cardano reveals Tartaglia’s secret solution
• Together solved general cubic and quartic!
Wednesday, April 18, 2012
Cardano and Ferrari
• Due to oath, could not publish their work!
• Traveled to Bologna seeking del Ferro’s
original work (1543)
• Found solution to depressed cubic!
• Cardano publishes Ars Magna in 1545
• Chapter XI “On the Cube and First Power
Equal to the Number”
Wednesday, April 18, 2012
Ars Magna
In our own days Scipione del Ferro of Bologna has
solved the case of the cube and first power equal to a
constant, a very elegant and admirable
accomplishment...In emulation of him, my friend
Niccolo Tartaglia of Brescia, wanting not to be
outdone, solved the same case when he got into a
contest with his [Scipione’s] pupil, Antonio Maria
Fior, and, moved by my many entreaties, gave it to me.
Wednesday, April 18, 2012
Ars Magna
For I had been deceived by the world of Luca Paccioli, who
denied that any more general rule could be discovered than
his own. Notwithstanding the many things which I had already
discovered, as is well known, I had despaired and had not
attempted to look any further. Then, however, having received
Tartaglia’s solution and seeking for the proof of it, I came to
understand that there were a great many other things that
could also be had. Pursuing this thought and with increased
confidence, I discovered these others, partly by myself and
partly through Lodovico Ferrari, formerly my pupil.
Wednesday, April 18, 2012
Ferrari vs. Tartaglia
• Public debate on August 10, 1548
• Refereed by Governor of Milan
• Each posed 62 problems
• Ferrari wins
There is a right-angled triangle, such that when the
perpendicular is drawn, one of the sides with the opposite
part of the base makes 30, and the other side with the other
part makes 28. What is the length of one of the sides?
Wednesday, April 18, 2012
Cardano’s Solution
Method to solve x3 + mx = n:
Cube one-third the coefficient of x; add to it the square
of one-half the constant of the equation; and take the
square root of the whole. You will duplicate [repeat]
this, and to one of the two you add one-half the number
you have already squared and from the other you
subtract one-half the same. Then, subtracting the cube
root of the first from the cube root of the second, the
remainder which is left is the value of x.
Wednesday, April 18, 2012
Cardano’s Solution
x =
�
Wednesday, April 18, 2012
3
n
+
2
�
n2 m3
+
−
4
27
�
3
−n
+
2
�
n2 m3
+
4
27
Cardano’s Solution
Vol of Pink Cube = u3
Vol of Green Cube = (t - u)3
Vol of Clear and Blue Slabs
= 2tu(t - u)
Vol of Yellow Block = u2(t - u)
Vol of Red Block = u(t - u)2
Total Volume (simplified):
t3 - u3 = (t - u)3 + 3tu(t - u)
Wednesday, April 18, 2012
Cardano’s Solution
Make a clever substitution in:
t3 - u3 = (t - u)3 + 3tu(t - u)
Let x = t - u to obtain:
x3 + 3tux = t3 - u3
This is depressed where m = 3tu and n = t3 - u3.
Solving for u in the first gives u = m/3t and
substituting this into the second gives:
n = t3 - m3/27t3
Wednesday, April 18, 2012
Cardano’s Solution
Multiplying n = t3 - m3/27t3 by t3 produces:
6
t -
3
nt
-
3
m /27
=0
which we can rewrite as:
3
2
(t ) -
3
n(t )
-
3
m /27
=0
This is a quadratic!!
Wednesday, April 18, 2012
Cardano’s Solution
(t3)2- n(t3) - m3/27 = 0
The quadratic formula gives solutions for t.
Then we use n = t3 - u3 to solve for u and finally
use x = t - u to solve for x. Thus, we have:
x =
Wednesday, April 18, 2012
�
3
n
+
2
�
n2 m3
+
−
4
27
�
3
−n
+
2
�
n2 m3
+
4
27
Ars Magna
Chapter XI: example illustrating
technique for x3 + 6x = 20
Chapter XII: solved x3 = mx + n
Chapter XIII: solved x3 + n = mx
But what about the general cubic:
3
2
ax + bx + cx + d = 0
Wednesday, April 18, 2012
General Cubic
ax3 + bx2 + cx + d = 0
The key is to make a clever substitution:
x = y - b/3a
This results in a depressed equation
y3 + py = q
Wednesday, April 18, 2012
Negative Roots
Puzzle: But what about negative roots?
Example: Find the roots of x3 - 15x = 4
x =
Wednesday, April 18, 2012
�
3
2+
√
−121 −
�
3
−2 +
√
−121
Negative Roots
(2 +
Rafael Bombelli
1526 - 1573
√
x =
√
√
3
−1) = 8 + 12 −1 − 6 − −1
√
= 2 + 11 −1
√
= 2 + −121
�
3
2+
√
−121 −
�
3
−2 +
√
−121
This image is from the website entry for Rafael Bombelli from the MacTutor History of Mathematics. It belongs to the public domain.
Wednesday, April 18, 2012
Negative Roots
Plus by plus of minus, makes plus of minus.
Minus by plus of minus, makes minus of minus.
Plus by minus of minus, makes minus of minus.
Minus by minus of minus, makes plus of minus.
Plus of minus by plus of minus, makes minus.
Plus of minus by minus of minus, makes plus.
Minus of minus by plus of minus, makes plus.
Minus of minus by minus of minus makes minus.
Rafael Bombelli
1526 - 1573
Wednesday, April 18, 2012
√
−x = “plus of minus”
Quartic Equation
Puzzle: What about the quartic?
4
3
2
ax + bx + cx + dx + e = 0
Step One: Divide by a and make a substitution
to obtain a depressed equation:
4
y
+
2
my
+ ny = p
Step Two: Replace this by a related cubic, then
use previous techniques.
Wednesday, April 18, 2012
Origami Solution
Elementary Moves:
Given two points P and Q, we can make a
crease line that places P onto Q when folded.
Given a line l and point P not on l, we can make
a crease line that passes through P and is
perpendicular to l.
Wednesday, April 18, 2012
rmation on these basic moves see [13] and [17]. Note, howev
sic moves can also be done by a straightedge and compass. The
which sets origami apart from straightedge and compass con
ng:
Beloch Fold
Fold. Given
Given two
two points
points PP1and
andPP1and
andtwo
twolines
linesl l1and
andl l2 we c
1
2
1
2
make a single fold that places P1 onto l1 and P2 onto l2 simu
we can, whenever possible, make a single fold that
.)
places P1 onto l1 and P2 onto l2 simultaneously.
P1
P2
l1
Figure 1. The Beloch origami fold.
Wednesday, April 18, 2012
l2
Beloch Fold
What is this fold accomplishing?
P
Wednesday, April 18, 2012
Beloch Fold
These crease lines are tangent to the parabola
with focus P and directrix l.
This image was created using the java applet from the website Cut The Knot.
Wednesday, April 18, 2012
Review of Parabolas
focus
directrix
This image is from the Wikipedia article on parabola. It belongs to the public domain.
Wednesday, April 18, 2012
Beloch Fold
Picture Proof:
P
The Beloch fold finds a common tangent
to two parabolas!
Wednesday, April 18, 2012
Beloch Fold
Morals:
Folding a point to a line is equivalent to solving a
quadratic equation.
The Beloch fold, then, is equivalent to solving a
cubic equation.
Wednesday, April 18, 2012
Margherita Piazzolla Beloch
• 1879 - 1976, Italian
• Algebraic geometer, Chair at Univ. of Ferrara
• First to discover origami can find common
tangents to two parabolas!
• Beloch fold is most complicated paper-folding
move possible
Wednesday, April 18, 2012
Margherita Piazzolla Beloch
Wednesday, April 18, 2012
Beloch Square
Beloch Square: Given two points A and B and
two lines r and s in the plane, construct a square
WXYZ with two adjacent corners X and Y lying
on r and s, respectively, and the sides WX and
YZ, or their extensions, passing through A and
B, respectively.
Wednesday, April 18, 2012
Beloch Square
Beloch Square: Given two points A and B and
�
a
line
s
left.) Note
that these line
two lines r and .in(See
the Figure
plane, 5,
construct
a square
viatwo
paper
foldingcorners
by, say, X
folding
WXYZ with
adjacent
and Yalong
lyingr , markin
then
making
a
sequence
of
perpendicular
folds
on r and s, respectively, and the sides WX and
this
are
left
as
an
exercise.)
YZ, or their extensions, passing through A and
B, respectively.
A
r
r�
B
s
s�
Wednesday, April 18, 2012
A
B
r
A
s
Beloch Square
Beloch Square: Given two points� A and� B and
(See Figure 5, left.) Note that these lines r and s can be constru
two lines r and s in the plane, construct a square
folding by, say, folding along r , marking where A lands under th
WXYZ with two adjacent corners X and Y lying
ing a sequence of perpendicular folds O2 described above. (The
on
r
and
s,
respectively,
and
the
sides
WX
and
ft as an exercise.)
YZ, or their extensions, passing through A and
B, respectively.
A
r
r�
B
s
s�
Wednesday, April 18, 2012
A
B
A
r
r�
A�
s
X
B
s�
B�
Y
r
Beloch Square
�
�
two pointseasily
A and B and
these Beloch
lines r Square:
and s canGiven
be constructed
two lines
r and
s in the
plane,
r , marking
where
A lands
under
thisconstruct
fold, anda square
WXYZ
two adjacent
corners
X andofY lying
ular folds
O2 with
described
above. (The
details
on r and s, respectively, and the sides WX and
YZ, or their extensions, passing through A and
B, respectively.
A
A
r
r�
A�
s
B
s�
B�
Wednesday, April 18, 2012
X
Y
r
r�
A�
s
s�
B�
Constructing the Cube Root of 2
r
A
s
B
Wednesday, April 18, 2012
Constructing the Cube Root of 2
r r’
s’
A
s
B
Wednesday, April 18, 2012
Constructing the Cube Root of 2
r r’
s’
A
s
B
Wednesday, April 18, 2012
Constructing the Cube Root of 2
r r’
A’
A
s’
s
B
Wednesday, April 18, 2012
B’
Constructing the Cube Root of 2
r r’
A’
X
A
Y
B
Wednesday, April 18, 2012
B’
s’
s
Constructing the Cube Root of 2
r r’
A’
X
A
Y
B
Wednesday, April 18, 2012
B’
s’
s
Constructing the Cube Root of 2
r r’
A’
X
A
Y
O
OAX
OXY
OBY
Wednesday, April 18, 2012
OX OY OB
=
=
OA OX OY
B
B’
s’
s
Constructing the Cube Root of 2
r r’
A’
X
A
Y
O
OAX
OXY
OBY
Wednesday, April 18, 2012
OX OY 2
= OX = OY
1
B
B’
s’
s
Constructing the Cube Root of 2
r r’
A’
X
A
Y
O
OY
2
.
.
(OX) = OX
OX OY
=2
3
Wednesday, April 18, 2012
B
B’
s’
s
Epilogue
Puzzle: What about the quintic?
Does there exist a “solution by radicals,” that is, a
formula for its roots that involves only the original
coefficients and the algebraic operations of
addition, subtraction, multiplication and division?
Wednesday, April 18, 2012
Epilogue
• 250 years since quartic solved
• 1790’s sends work to Lagrange
Paolo Ruffini
1765 - 1822
The algebraic solution of general equations of
degree greater than four is always impossible.
Behold a very important theorem which I
believe I am able to assert (if I do not err): to
present the proof of it is the main reason for
publishing this volume. The immortal
Lagrange, with his sublime reflections, has
provided the basis of my proof.
This image is from the Wikipedia article on Paolo Ruffini. It belongs to the public domain.
Wednesday, April 18, 2012
Epilogue
• 250 years since quartic solved
• 1790’s sends work to Lagrange
• Sends work to Institute of Paris
and Royal Society
Paolo Ruffini
1765 - 1822
... if a thing is not of importance, no
notice is taken of it and Lagrange
himself, “with his coolness” found
little in it worthy of attention.
This image is from the Wikipedia article on Paolo Ruffini. It belongs to the public domain.
Wednesday, April 18, 2012
Epilogue
Geometers have occupied themselves
a great deal with the general
solution of algebraic equations and
several among them have sought to
prove the impossibility. But, if I am
not mistaken, they have not
succeeded up to the present. (1824)
Niels Abel
1802 - 1829
Why does Abel get the credit?
This image is from the Wikipedia article on Niels Henrik Abel. It belongs to the public domain.
Wednesday, April 18, 2012
Epilogue
... the mathematical community was not ready to
accept so revolutionary an idea: that a polynomial
could not be solved in radicals. Then, too, the
method of permutations was too exotic and, it
must be conceded, Ruffini's early account is not
easy to follow. ... between 1800 and 1820 say, the
mood of the mathematical community ... changed
from one attempting to solve the quintic to one
proving its impossibility...
Wednesday, April 18, 2012
References
• Dunham, W. Journey through Genius:
The
Great Theorems of Mathematics, John Wiley &
Sons: New York, 1990, 133 - 154
• Hull, T. “Solving Cubics with Creases:
The
Work of Beloch and Lill,” American
Mathematical Monthly Vol. 118, No. 4 (April
2011), 307 - 315
• MacTutor History of Mathematics
Wednesday, April 18, 2012